Problem: Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$.  Each of these three parts of the trip is made entirely by bus or entirely by airplane.  The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a bus, it costs Piravena $\$0.15$ per kilometer. To take an airplane, it costs her a $\$100$ booking fee, plus $\$0.10$ per kilometer. [asy]

pair A, B, C;

C=(0,0);

B=(0,1250);

A=(3000,0);

draw(A--B--C--A);

label("A", A, SE);

label("B", B, NW);

label("C", C, SW);

label("3000 km", (A+C)/2, S);

label("3250 km", (A+B)/2, NE);

draw((0,125)--(125,125)--(125,0));

[/asy] Determine the distance she travels for her complete trip.
Explanation: Since $\triangle ABC$ is a right-angled triangle, then we may use the Pythagorean Theorem.
Thus, $AB^2=BC^2+CA^2$, and so  \begin{align*}
BC^2&=AB^2-CA^2\\
&=3250^2-3000^2\\
&=250^2(13^2-12^2)\\
&=250^2(5^2)\\
&=1250^2.
\end{align*} therefore $BC=1250$ km (since $BC>0$).
Piravena travels a distance of $3250+1250+3000=\boxed{7500}$ km for her complete trip.